Numerical Computation of Geodesics
on Combined Piecewise Smooth
Surfaces
Ron Caplan, Michael Davis, Jenny Dehner, Nick Hooper, Heather Mattie
Advisor: Ali Nadim
Boeing Liaison: Tom Grandine
Claremont Graduate University
Boeing Clinic 08-09
Claremont Graduate University
• 2188 graduate
students
• A member of the
Claremont Colleges
– Pomona College
– Harvey Mudd
College
– Claremont
McKenna College
– Etc.
• School of Mathematical
Sciences
– Mathematics Clinic
The Proposed Problem
• Boeing 787
– composite tape
– must be laid flat
– no wrinkles, creases
• Laying each strip along
a geodesic minimizes
creasing and wrinkling
http://en.wikipedia.org/wiki/File:Boeing_787.jpg
• Need an algorithm that can
aid in the process of finding
geodesics along the combined
surfaces that make up a
fuselage
Overview
• Foundational work
• The main algorithm
• A few test runs
• Challenges and improvements
• Conclusions
Getting Acquainted with
Geodesics
• How we got started:
– Reviewed previous work from CGU Boeing
Clinic Team 07-08
– Performed literature review
– Reviewed topics in differential geometry
– “Evolution” algorithm seemed best path to
follow
Previous Team’s Work
• Dijkstra’s Algorithm
– Limited to points and edges on the mesh
• Fast Marching Method and Level Set Method
– Difficult to program and run
• Modified Dijkstra’s Algorithm
– Does not take the
curvature of the surface
into account
– Can only be used on
meshed surfaces
Toy Problem: A Cylinder
• Our first simple
introduction to shortest
paths and geodesics
– Compute shortest paths
analytically using
distance formulas
– Gives us a reference for
methods developed later
Our New Method
• An algorithm for evolution of an initial
curve toward the geodesic
• Achieved by computing numerical
properties of the curve and moving
marker points along the curve in
proportion to the geodesic curvature
vector
The Evolution Algorithm
• Given some 3-D surface parameterized by
by the equation:
represented
the initial point
and the end point
, connect the
two points in 2-dimensional space with , some smooth initial
curve. (Many times, a straight line between the points is the
easiest.)
The Evolution Algorithm
• Discretize the smooth curve , parameterized by
arclength, into N points such that
refers to the
point in the list of points and map these points onto the
surface.
• Jkfkxcjvbxckjbvxv
• Calculate arclength at each point in the
on the 3-D surface.
plane and
The Evolution Algorithm
• In 3-dimensional space, calculate the normal
the surface at every point on the curve.
to
Also in 3-dimensional space, calculate the
curvature vector:
The Evolution Algorithm
• From the previous steps, calculate the
geodesic curvature
at every point.
• Geodesic occurs when:
• Stopping Criterion:
– Check sum of squares
of 2-norm. If it is larger
than some tolerance, take
a evolution step.
The Evolution Algorithm
• Compute the tangent vector to the curve at each point
• Calculate a signed scalar
quantity at each point that
represents a velocity with
which each point will move
• At every point along the
curve in the
plane
calculate a local normal
to the curve
The Evolution Algorithm
• Move each point in the
plane in the direction of the
normal vector to the curve and proportional to the
calculated velocity
The Evolution Algorithm
• Repeat previous steps until the stopping
criterion (
) is satisfied:
Implementation
• Derivatives Subroutine (CGUBC_deriv)
– Central Difference Method
• Calculates first derivatives quickly
• Computes second derivatives with low
accuracy
– Cubic Spline Method
• Numerically calculates more accurate
second derivatives
• Increases complexity of subroutine
Implementation
• Time Integration:
• 4th Order Runga-Kutta:
• Time step needs to be chosen carefully
Example #1
• Surface: Cylinder
• Initial condition near geodesic
Example #2
• Surface: Cylinder
• Complicated initial condition
• We see clumping of points and instability “blip”
Example #3a
• Surface: Cylinder
• We see boundary condition enforcement
• Clumping, but no blip.
Example #3b
• Surface: Cylinder
• Using cubic spline derivatives
– Less clumping
Example #3c
• Surface: Cylinder
• Using u-redistribution
– No clumping
Example #4
• Surface: Sphere
• More sensitive
• Still works well for “nice” initial curves away from pole
Challenges & Solutions
• Challenges
– Convergence rate slow for some problems
– Stability requirements on time step
– “Singular” points cause breakdown
• Possible solutions
– Implicit time scheme
– Manual/automatic singular point
computations
Conclusions / Results
• Current evolution process can be used to
relax an initial curve toward a geodesic
• Some issues remain to be addressed
• In principle, the algorithm should be able
to handle constraints (with properly
modified stopping criterion) and multiple
surfaces
Future Work
• Constraints on surfaces
– Include regions that are inaccessible
• Multiple surfaces
– Crossing edges
• Other test cases
– Objects with corner
Acknowledgements
Tom Grandine, Boeing Liaison
Ali Nadim, CGU faculty mentor
This material is based upon work supported by the National
Science Foundation under Grant No. 0538663.
Any opinions, findings, and conclusions or recommendations
expressed in this material are those of the authors and do not
necessarily reflect the views of the National Science Foundation.
References
[1] S.V. Fomin I.M. Gelfand. Calculus of Variations. Dover Publications, Inc., 2000.
[2] C.H. Papadimitrious J.S.B. Mitchell, D.M. Mount. The discrete geodesic
problem. SIAM Journal on Computing, 16(4):647–668, 1987.
[3] M. Schmies K. Polithier. Straightest geodesics on polyhedral surfaces. pages 30–38,
2006.
[4] E.J. Candes L. Ying. Fast geodesics computation with the phase flow method. Journal
of Computational Physics, 220(1):6–18, 2006.
[5] E.J. Candes L. Ying. The phase flow method. Journal of Computational
Physics, 220(1):184–215, 2006.
[6] J.A. Sethian R. Kimmel. Computing geodesic paths on manifolds. Proceedings of
the National Academy of Sciences of the USA, 95(15):8431–8435,
1998.
[7] D. Breen S. Mauch. A fast marching method of computing closest points.
[8] V. Toponogov. Differential geometry of curves and surfaces: a concise guide.
Springer, 2006.
[9] D. Kirsanov S.J. Gortler H. Hoppe V. Surazhsky, T. Sur. Fast exact and
approximate geodesics on meshes. ACM Trans. Graph, 24(3):553–560, 2005.
[10] S.-M. Hu Y.-J. Liu, Q.-Y. Zhou. Handling degenerate cases in exact
computation on triangle meshes. The Visual Computer, 23(9):661–668,
September 2007.
[11] D. Zorin. Curvature and geodesics, discrete laplacian and related smoothing
methods. pages 1–6, 2002.